The main subjects of the paper is studying the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups $��_1(M)$ of symplectically aspherical manifolds $M$ with $��_2(M)=0$ and $��_2(M)\neq 0$. Relations between these classes are discussed. We show that several important classes of groups can be realized in both classes, while some of groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general research of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical, etc.

24 pages; the paper has a new structure, new results and more references; For example, we have added a new section "Symplectic bundles and symplectic asphericity" and also new results on the section "Gompf symplectic sum and symplectic asphericity"